Lecture 37.

We noted that we now have all the necessary elements to build quantum
field theories: we know the free field theories for scalar, spinor,
and 4-vector fields; we know how the free field quanta (particles and
antiparticles) propagate; and we know how to include interactions
(potential terms), as point interactions that create and annihilate
groups of particles. We now put these elements together to build the
quantum field theories of the particles and interactions in our world.
We need only decide 1) how many particles of which types to include;
and 2) which interaction terms to allow.

To make these choices, we are guided by symmetry principles. We
reviewed the observable consequences of symmetry that we have seen
already in this course: local conservation laws for the generators of
the symmetries; and degeneracies of the allowed energies (or particle
masses). (For partial symmetry, the degeneracies can lift to give
patterns in the allowed energies). We also noted that these observable
consequences correlate to required symmetries of the Lagrangian:
having observed the consequences of a given symmetry, we can
include in our Lagrangian only the interaction terms which are invariant
(up to a total derivative) under that symmetry.

We noted that symmetries come in 2 types: *global*, which we have
studied so far, and *gauged*. We defined a gauged symmetry as one
where the symmetry transformation --- or change of basis --- could be
chosen independently at all points in spacetime, *still* without
affecting the observed physics. We argued certain theorist's reasons
as to why this should still be a symmetry: why other observer's basis
choices at far-removed points should not affect our local observables,
in a simple and just world.

We then went about building a theory with a gauged symmetry, focusing
on gauging the phase symmetry of the Dirac Lagrangian. We first
considered potential terms in the Lagrangian, which depend only on
\psi and \bar{\psi}, and not on derivatives. These, if they
are invariant under the original global phase symmetry, are
automatically invariant under the gauged (that is, locally varying)
phase symmetry.  Our only problem arises from the gradient terms in
the Lagrangian, as the gradient of our phase choice contributes a
non-invariant term.

We set about introducing a gauge field A_\mu, whose sole purpose was
to cancel the noninvariant contribution due to \del_\mu \lambda
(where \lambda is the phase). We found that we could do this by
systematically replacing all gradient terms \del_\mu  \psi
with a ``covariant'' gradient D_\mu  \psi = (\del_\mu -ie
A_\mu) \psi, with A_\mu transforming under the gauged phase
transformation as A_\mu goes to A_\mu - e^{-1} \del_\mu
\lambda. This caused the term D_\mu \psi to transform in the exact
same way as \psi, so that it automatically appeared in the
Lagrangian only in the proper invariant combinations (since the
Lagrangian had been set up to compensate for the expected
transformation properties of \psi).

Replacing all derivatives by covariant derivatives is called ``minimal
coupling''. Note that this automatically produced, for the Dirac
Lagrangian, an interaction coupling \psi, \bar{\psi}, and the gauge
field A_\mu. We noted also that the required transformation law for
A^\mu was exactly the ``gauge transformation'' we remember from
electromagnetism. Thus we can identify A^\mu with the photon field,
and build the gauge-invariant kinetic term - F_{\mu\nu}F^{\mu\nu}/4,
with our usual F_{\mu\nu} = \del_\mu A_\nu - \del_\nu A_\mu. We also
identify the conserved current j^\mu associated with the phase
transformation as the electromagnetic current. We concluded then that
our theory, built from symmetry considerations, is just
electromagnetism, of the charged particles e^+ and e^- and the photon
A^\mu.

We discussed the predicted interaction vertices between electrons,
positrons, and photons. We noted that this theory --- Quantum
ElectroDynamics, QED --- had great success and was the early stimulus
of interest in quantum field theory. We also noted that, in the
classical limit, the minimal coupling prescription for the gradient
becomes the prescription mv - eA for the electron's conjugate
momentum, which reproduces the Lorentz force law for an electrically
charged particle.


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